correction of states Enzo
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@ -198,15 +198,17 @@
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\sisetup{range-units=single}
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%states
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\newcommand{\Ag}{$2{}^1A_g$}
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\newcommand{\Aoneg}{$2{}^1A_{1g}$}
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\newcommand{\Btwog}{$1{}^1B_{2g}$}
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\newcommand{\Atwog}{$1{}^3A_{2g}$}
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%D2h states
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\newcommand{\oneAg}{$1{}^1A_g$}
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\newcommand{\tBoneg}{$1{}^3B_{1g}$}
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\newcommand{\sBoneg}{$1{}^1B_{1g}$}
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\newcommand{\twoAg}{$2{}^1A_g$}
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%D4h states
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%\newcommand{\oneBoneg}{$1{}^1B_{1g}$} same label as the D2h state
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\newcommand{\Atwog}{$1{}^3A_{2g}$}
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\newcommand{\Aoneg}{$2{}^1A_{1g}$}
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\newcommand{\Btwog}{$1{}^1B_{2g}$}
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\begin{document}
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@ -254,7 +256,7 @@ This degeneracy is lifted by the so-called Jahn-Teller effect, \ie, by a descent
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In such as case, H\"uckel molecular orbital theory (correctly) predicts a closed-shell singlet ground state at the {\Dtwo} rectangular geometry.
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\titou{Indeed, synthetic work from Pettis and co-workers \cite{reeves_1969} gives a rectangular geometry to the singlet ground state of CBD and then was confirmed by experimental works. \cite{irngartinger_1983,ermer_1983,kreile_1986}}
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At the {\Dtwo} ground-state structure, the \titou{{\Ag}} state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods.
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At the {\Dtwo} ground-state structure, the \titou{{\oneAg}} state has a weak multi-configurational character with well-separated frontier orbitals that can be described by single-reference methods.
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However, at the {\Dfour} square geometry, the singlet state {\sBoneg} has two singly occupied frontier orbitals that are degenerated so has a two-configurational character and single-reference methods are unreliable to describe it.
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The singlet (\Dfour) is a transition state in the automerization reaction between the two rectangular structures (see Fig.~\ref{fig:CBD}).
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The autoisomerization barrier (AB) for the CBD molecule is defined as the energy difference between the singlet ground state of the square (\Dfour) structure and the singlet ground state of the rectangular (\Dtwo) geometry.
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@ -262,9 +264,8 @@ The energy of this barrier was predicted, experimentally, in the range of 1.6-10
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All the specificities of CBD make it a real playground for excited-states methods.
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The lowest-energy excited states of CBD in both geometries are represented in Fig.~\ref{fig:CBD}.
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Are represented {\Ag} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one.
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Due to energy scaling doubly excited state {\sBoneg} and {\Aoneg} for the {\Dtwo} and {\Dfour} structures, respectively, are not drawn.
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Doubly excited states are known to be challenging to represent for adiabatic time-dependent density functional theory \cite{casida_1995} (TD-DFT) and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{christiansen_1995,koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{kucharski_1991,kallay_2004,hirata_2000,hirata_2004}
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Are represented {\oneAg} and {\tBoneg} states for the rectangular geometry and {\sBoneg} and {\Atwog} for the square one.
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Due to energy scaling, the {\sBoneg}, {\twoAg} and {\Aoneg}, {\Btwog}, states for the {\Dtwo} and {\Dfour} structures, respectively, are not drawn. The {\twoAg} and {\Aoneg} states are doubly excited states and these doubly excited states are known to be challenging to represent for adiabatic time-dependent density functional theory \cite{casida_1995} (TD-DFT) and even for state-of-the-art methods like the approximate third-order coupled-cluster (CC3) \cite{christiansen_1995,koch_1997} or equation-of-motion coupled-cluster with singles, doubles and triples (EOM-CCSDT). \cite{kucharski_1991,kallay_2004,hirata_2000,hirata_2004}
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In order to tackle the problems of multi-configurational character and double excitations several ways are explored. The most evident way that one can think about to describe multiconfigurational and double excitations are multiconfigurational methods.
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Among these methods, one can find complete active space self-consistent field (CASSCF) \cite{roos_1996}, the second perturbation-corrected variant (CASPT2) \cite{andersson_1990} and the second-order $n$-electron valence state perturbation theory (NEVPT2). \cite{angeli_2001,angeli_2001a,angeli_2002}
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@ -283,12 +284,12 @@ So one can access the singlet ground state and the singlet doubly-excited state
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Obviously spin-flip methods have their own flaws, especially the spin contamination \cite{casanova_2020} (\ie, an artificial mixing of electronic states of different spin multiplicities) due to spin incompleteness of the spin-flip expansion and by spin contamination of the reference configuration.
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One can address part of this problem by expansion of the excitation order but with an increase of the computational cost or by complementing the spin-incomplete configuration set with the missing configurations.
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In the present work we investigate $1{}^1A_{g}$, {\tBoneg}, {\sBoneg}, {\Ag} and {\sBoneg}, {\Atwog}, {\Aoneg},{\Btwog} excited states for the {\Dtwo} and {\Dfour} geometries, respectively.
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In the present work we investigate {\oneAg}, {\tBoneg}, {\sBoneg}, {\twoAg} and {\sBoneg}, {\Atwog}, {\Aoneg},{\Btwog} excited states for the {\Dtwo} and {\Dfour} geometries, respectively.
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Computational details are reported in Section \ref{sec:compmet} for SCI (Subsection \ref{sec:SCI}), EOM-CC (Subsection \ref{sec:CC}), multiconfigurational (Subsection \ref{sec:Multi}) and spin-flip (Subsection \ref{sec:sf}) methods.
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Section \ref{sec:res} is devoted to the discussion of our results, first we consider the ground state property studied which is the AB (Subsection \ref{sec:auto}) and then we study the excited states (Subsection \ref{sec:states}) of the CBD molecule for both geometries.
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\begin{figure}
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\includegraphics[width=0.6\linewidth]{figure2.png}
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\includegraphics[width=0.6\linewidth]{figure1.pdf}
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\caption{Pictorial representation of the ground and excited states of CBD and its properties under investigation.
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The singlet ground-state (S) and triplet (T) properties are represented in black and red, respectively.}
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\label{fig:CBD}
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@ -372,13 +373,13 @@ Two different sets of geometries obtained with different level of theory are con
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\begin{tabular}{lllrrr}
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State & Method & \ce{C=C} & \ce{C-C} & \ce{C-H} & \ce{H-C=C}\fnm[1] \\
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\hline
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{\Dtwo} $({}^1 A_{g})$&
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{\Dtwo} ({\oneAg}) &
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CASPT2(12,12)/aug-cc-pVTZ & 1.355 & 1.566 & 1.077 & 134.99 \\
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&CC3/aug-cc-pVTZ & 1.344 & 1.565 & 1.076 & 135.08 \\
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&CCSD(T)/cc-pVTZ & 1.343 & 1.566 & 1.074 & 135.09 \fnm[2]\\
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{\Dfour} $({}^1 B_{1g})$&
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{\Dfour} ({\sBoneg}) &
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CASPT2(12,12)/aug-cc-pVTZ & 1.449 & 1.449 & 1.076 & 135.00 \\
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{\Dfour} $({}^3 A_{2g})$&
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{\Dfour} ({\Atwog}) &
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CASPT2(12,12)/aug-cc-pVTZ & 1.445 & 1.445 & 1.076 & 135.00 \\
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&RO-CCSD(T)/aug-cc-pVTZ & 1.439 & 1.439 & 1.075 & 135.00
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\end{tabular}
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@ -463,24 +464,24 @@ Figure \ref{fig:AB} shows the autoisomerization barrier (AB) energies for the CB
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%All the calculations are performed using four basis set, the 6-31+G(d) basis and the aug-cc-pVXZ with X$=$D, T, Q. In the following we will use the notation AVXZ for the aug-cc-pVXZ basis, again with X$=$D, T, Q.
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\subsubsection{{\Dtwo} geometry}
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Table \ref{tab:sf_tddft_D2h} shows the results obtained for the vertical transition energies using spin-flip methods and Table \ref{tab:D2h} shows the results obtained with the standard methods. We discuss first the SF-TD-DFT values with hybrid functionals. For the B3LYP functional we can see that the energy differences for each state and throughout all bases are small with the largest one for the {\tBoneg} state with \SI{0.012}{\eV}. The same observation can be done for the PBE0 and BH\&HLYP functionals, we can also observe that adding exact exchange to the functional (20\% of exact exchange for B3LYP, 25\% for PBE0 and 50\% for BH\&HLYP increase the energy difference between the {\sBoneg} and the {\tBoneg} states. Put another way, increasing the amount of exact exchange in the functional reduces the energy difference between the {\tBoneg} and the $\text{X}\,{}^1A_{g}$ states. We can also notice that the vertical energies of the different states do not vary in the same way when adding exact exchange, for instance the energy variation for the {\tBoneg} state from PBE0 to BH\&HLYP is around \SI{0.1}{\eV} whereas for the {\sBoneg} and the {\Ag} states this energy variation is about \SIrange{0.4}{0.5}{\eV} and \SI{0.34}{eV} respectively. For the RSH functionals we can not make the same observation, indeed we can see that for the CAM-B3LYP functional we have that the energy difference between the {\sBoneg} and the {\tBoneg} states is larger than for the $\omega$B97X-V and LC-$\omega $PBE08 functionals despite the fact that the latter ones have a bigger amount of exact exchange. However we can observe that we have a small energy difference for each state throughout the bases and for each RSH functional. The M06-2X functional is an Hybrid meta-GGA functional and contains 54\% of Hartree-Fock exchange, we can compare the energies with the BH\&HLYP functional and we can see that the energy differences are small with around \SIrange{0.03}{0.08}{\eV}. We can notice that the upper bound of \SI{0.08}{\eV} in the energy differences is due to the {\tBoneg} state. The M11 vertical energies are close to the BH\&HLYP ones for the triplet and the first singlet excited states with \SIrange{0.01}{0.02}{\eV} and \SIrange{0.08}{0.10}{\eV} of energy difference, respectively. For the {\Ag} state the M11 energies are closer to the $\omega$B97X-V ones with \SIrange{0.05}{0.09}{\eV} of energy difference.
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Table \ref{tab:sf_tddft_D2h} shows the results obtained for the vertical transition energies using spin-flip methods and Table \ref{tab:D2h} shows the results obtained with the standard methods. We discuss first the SF-TD-DFT values with hybrid functionals. For the B3LYP functional we can see that the energy differences for each state and throughout all bases are small with the largest one for the {\tBoneg} state with \SI{0.012}{\eV}. The same observation can be done for the PBE0 and BH\&HLYP functionals, we can also observe that adding exact exchange to the functional (20\% of exact exchange for B3LYP, 25\% for PBE0 and 50\% for BH\&HLYP increase the energy difference between the {\sBoneg} and the {\tBoneg} states. Put another way, increasing the amount of exact exchange in the functional reduces the energy difference between the {\tBoneg} and the {\oneAg} states. We can also notice that the vertical energies of the different states do not vary in the same way when adding exact exchange, for instance the energy variation for the {\tBoneg} state from PBE0 to BH\&HLYP is around \SI{0.1}{\eV} whereas for the {\sBoneg} and the {\twoAg} states this energy variation is about \SIrange{0.4}{0.5}{\eV} and \SI{0.34}{eV} respectively. For the RSH functionals we can not make the same observation, indeed we can see that for the CAM-B3LYP functional we have that the energy difference between the {\sBoneg} and the {\tBoneg} states is larger than for the $\omega$B97X-V and LC-$\omega $PBE08 functionals despite the fact that the latter ones have a bigger amount of exact exchange. However we can observe that we have a small energy difference for each state throughout the bases and for each RSH functional. The M06-2X functional is an Hybrid meta-GGA functional and contains 54\% of Hartree-Fock exchange, we can compare the energies with the BH\&HLYP functional and we can see that the energy differences are small with around \SIrange{0.03}{0.08}{\eV}. We can notice that the upper bound of \SI{0.08}{\eV} in the energy differences is due to the {\tBoneg} state. The M11 vertical energies are close to the BH\&HLYP ones for the triplet and the first singlet excited states with \SIrange{0.01}{0.02}{\eV} and \SIrange{0.08}{0.10}{\eV} of energy difference, respectively. For the {\twoAg} state the M11 energies are closer to the $\omega$B97X-V ones with \SIrange{0.05}{0.09}{\eV} of energy difference.
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Then we discuss the various ADC scheme (ADC(2)-s, ADC(2)-x and ADC(3)) results. For ADC(2) we have vertical energy differences of about 0.03 eV for the {\tBoneg} state and around 0.06 eV for {\Ag} state throughout all bases. However for the {\sBoneg} state we have an energy difference of about 0.2 eV. For the ADC(2)-x and ADC(3) schemes the calculations with the aug-cc-pVQZ basis are not feasible with our resources. With ADC(2)-x we have similar vertical energies for the triplet and the {\sBoneg} states but for the {\Ag} state the energy difference between the ADC(2) and ADC(2)-x schemes is about \SIrange{0.4}{0.5}{\eV}. The ADC(3) values are closer from the ADC(2) than the ADC(2)-x except for the triplet state for which we have a energy difference of about \SIrange{0.09}{0.14}{\eV}. Now, we look at Table \ref{tab:D2h} to discuss the results of standard methods. First we discuss the CC values, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. We can notice that for the {\tBoneg} and the {\sBoneg} states the CCSDT and the CC3 values are close with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases. The energy difference is larger for the {\Aoneg} state with around \SIrange{0.35}{0.38}{\eV}. The same observation can be done for CCSDTQ and CC4 with similar vertical energies for all bases. Note that the {\tBoneg} state can not be described with these methods.
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Then we discuss the various ADC scheme (ADC(2)-s, ADC(2)-x and ADC(3)) results. For ADC(2) we have vertical energy differences of about 0.03 eV for the {\tBoneg} state and around 0.06 eV for {\twoAg} state throughout all bases. However for the {\sBoneg} state we have an energy difference of about 0.2 eV. For the ADC(2)-x and ADC(3) schemes the calculations with the aug-cc-pVQZ basis are not feasible with our resources. With ADC(2)-x we have similar vertical energies for the triplet and the {\sBoneg} states but for the {\twoAg} state the energy difference between the ADC(2) and ADC(2)-x schemes is about \SIrange{0.4}{0.5}{\eV}. The ADC(3) values are closer from the ADC(2) than the ADC(2)-x except for the triplet state for which we have a energy difference of about \SIrange{0.09}{0.14}{\eV}. Now, we look at Table \ref{tab:D2h} to discuss the results of standard methods. First we discuss the CC values, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. We can notice that for the {\tBoneg} and the {\sBoneg} states the CCSDT and the CC3 values are close with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases. The energy difference is larger for the {\twoAg} state with around \SIrange{0.35}{0.38}{\eV}. The same observation can be done for CCSDTQ and CC4 with similar vertical energies for all bases. Note that the {\tBoneg} state can not be described with these methods.
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Then we review the vertical energies obtained with multireference methods. The smallest active space considered is four electrons in four orbitals, for CASSCF(4,4) we have small energy variations throughout bases for the {\tBoneg} and the {\Ag} states but a larger variation for the {\sBoneg} state with around \SI{0.1}{\eV}. We can observe that we have the inversion of the states compared to all methods discussed so far between the {\Ag} and {\sBoneg} states with {\sBoneg} higher than {\Ag} due to the lack of dynamical correlation in the CASSCF methods. The {\sBoneg} state values in CASSCF(4,4) is much higher than for any of the other methods discussed so far. With CASPT2(4,4) we retrieve the right ordering between the states and we see large energy differences with the CASSCF values. Indeed, we have approximatively \SIrange{0.22}{0.25}{\eV} of energy difference for the triplet state for all bases and \SIrange{0.32}{0.36}{\eV} for the {\Ag} state, the largest energy difference is for the {\sBoneg} state with \SIrange{1.5}{1.6}{\eV}.
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For the XMS-CASPT2(4,4) only the {\Ag} state is described with values similar than for the CAPST2(4,4) method. For the NEVPT2(4,4) methods (SC-NEVPT2 and PC-NEVPT2) the vertical energies are similar for the {\sBoneg} and the {\Ag} states with approximatively \SIrange{0.002}{0.003}{\eV} and \SIrange{0.02}{0.03}{\eV} of energy difference for all bases, respectively. The energy difference for the {\sBoneg} state is slightly larger with \SI{0.05}{\eV} for all bases. Note that for this state the vertical energy varies of \SI{0.23}{eV} from the 6-31+G(d) basis to the aug-cc-pVDZ one. Then we use a larger active space with twelve electrons in twelve orbitals, the CASSF(12,12) values are close to the CASSCF(4,4) value for the triplet state with 0.01-0.02 eV of energy differences. For the {\Ag} state we have an energy difference of about \SI{0.2}{eV} between the CASSCF(4,4) and the CASSCF(12,12) values. We can notice that increasing the size of the active space gives the right ordering for the states and we have an energy difference of around \SI{0.7}{\eV} for the {\sBoneg} state between CASSCF(4,4) and the CASSCF(12,12) values. The CASPT2(12,12) method decreases the energy of all states compared to the CASSCF(12,12) method, again the decrease is not the same for all states. We have a diminution from CASSCF to CASPT2 of about \SIrange{0.17}{0.2}{\eV} for the {\tBoneg} and the {\Ag} states and for the different bases. Again, the energy difference for the {\Ag} state is larger with \SIrange{0.5}{0.7}{\eV} depending on the basis. In a similar way than with XMS-CASPT2(4,4), XMS-CASPT(12,12) only describes the {\Ag} state and the vertical energies for this state are close to the CASPT(12,12) values. For the NEVPT2(12,12) schemes we see that for the triplet and the {\Aoneg} states the energies are similar with an energy difference between the SC-NEVPT2(12,12) and the PC-NEVPT2(12,12) values of about \SIrange{0.03}{0.04}{\eV} and \SIrange{0.02}{0.03}{\eV} respectively.
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Then we review the vertical energies obtained with multireference methods. The smallest active space considered is four electrons in four orbitals, for CASSCF(4,4) we have small energy variations throughout bases for the {\tBoneg} and the {\twoAg} states but a larger variation for the {\sBoneg} state with around \SI{0.1}{\eV}. We can observe that we have the inversion of the states compared to all methods discussed so far between the {\twoAg} and {\sBoneg} states with {\sBoneg} higher than {\twoAg} due to the lack of dynamical correlation in the CASSCF methods. The {\sBoneg} state values in CASSCF(4,4) is much higher than for any of the other methods discussed so far. With CASPT2(4,4) we retrieve the right ordering between the states and we see large energy differences with the CASSCF values. Indeed, we have approximatively \SIrange{0.22}{0.25}{\eV} of energy difference for the triplet state for all bases and \SIrange{0.32}{0.36}{\eV} for the {\twoAg} state, the largest energy difference is for the {\sBoneg} state with \SIrange{1.5}{1.6}{\eV}.
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For the XMS-CASPT2(4,4) only the {\twoAg} state is described with values similar than for the CAPST2(4,4) method. For the NEVPT2(4,4) methods (SC-NEVPT2 and PC-NEVPT2) the vertical energies are similar for the {\sBoneg} and the {\twoAg} states with approximatively \SIrange{0.002}{0.003}{\eV} and \SIrange{0.02}{0.03}{\eV} of energy difference for all bases, respectively. The energy difference for the {\sBoneg} state is slightly larger with \SI{0.05}{\eV} for all bases. Note that for this state the vertical energy varies of \SI{0.23}{eV} from the 6-31+G(d) basis to the aug-cc-pVDZ one. Then we use a larger active space with twelve electrons in twelve orbitals, the CASSF(12,12) values are close to the CASSCF(4,4) value for the triplet state with 0.01-0.02 eV of energy differences. For the {\twoAg} state we have an energy difference of about \SI{0.2}{eV} between the CASSCF(4,4) and the CASSCF(12,12) values. We can notice that increasing the size of the active space gives the right ordering for the states and we have an energy difference of around \SI{0.7}{\eV} for the {\sBoneg} state between CASSCF(4,4) and the CASSCF(12,12) values. The CASPT2(12,12) method decreases the energy of all states compared to the CASSCF(12,12) method, again the decrease is not the same for all states. We have a diminution from CASSCF to CASPT2 of about \SIrange{0.17}{0.2}{\eV} for the {\tBoneg} and the {\twoAg} states and for the different bases. Again, the energy difference for the {\twoAg} state is larger with \SIrange{0.5}{0.7}{\eV} depending on the basis. In a similar way than with XMS-CASPT2(4,4), XMS-CASPT(12,12) only describes the {\twoAg} state and the vertical energies for this state are close to the CASPT(12,12) values. For the NEVPT2(12,12) schemes we see that for the triplet and the {\twoAg} states the energies are similar with an energy difference between the SC-NEVPT2(12,12) and the PC-NEVPT2(12,12) values of about \SIrange{0.03}{0.04}{\eV} and \SIrange{0.02}{0.03}{\eV} respectively.
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%%% TABLE II %%%
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\begin{squeezetable}
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\begin{table}
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\caption{
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Spin-flip TD-DFT vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the {\tBoneg}, {\sBoneg}, and {\Ag} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
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Spin-flip TD-DFT vertical excitation energies (with respect to the singlet {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.
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\label{tab:sf_tddft_D2h}}
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\begin{ruledtabular}
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\begin{tabular}{llrrr}
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& \mc{4}{r}{Excitation energies (eV)} \hspace{0.5cm}\\
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\cline{3-5}
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Method & Basis & {\tBoneg} & {\sBoneg} & {\Ag} \\
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Method & Basis & {\tBoneg} & {\sBoneg} & {\twoAg} \\
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\hline
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% SF-TD-BLYP & 6-31+G(d) & $1.829$ & $1.926$ & $3.755$ \\
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% & aug-cc-pVDZ & $1.828$ & $1.927$ & $3.586$ \\
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@ -549,13 +550,13 @@ SF-ADC(3) & 6-31+G(d) & $1.435$ & $3.352$ & $4.242$ \\
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\begin{squeezetable}
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\begin{table}
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\caption{
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Vertical excitation energies (with respect to the singlet $\text{X}\,{}^1A_{g}$ ground state) of the {\tBoneg}, {\sBoneg}, and{\Ag} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the $\text{X}\,{}^1 A_{g}$ ground state.
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Vertical excitation energies (with respect to the singlet {\oneAg} ground state) of the {\tBoneg}, {\sBoneg}, and {\twoAg} states of CBD at the {\Dtwo} rectangular equilibrium geometry of the {\oneAg} ground state.
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\label{tab:D2h}}
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\begin{ruledtabular}
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\begin{tabular}{llrrr}
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& & \mc{3}{c}{Excitation energies (eV)} \\
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\cline{3-5}
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Method & Basis & {\tBoneg} & {\sBoneg} & {\Ag} \\
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Method & Basis & {\tBoneg} & {\sBoneg} & {\twoAg} \\
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\hline
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CC3 &6-31+G(d)& $1.420$ & $3.341$ & $4.658$ \\
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& aug-cc-pVDZ & $1.396$ & $3.158$ & $4.711$ \\
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@ -627,13 +628,13 @@ CIPSI &6-31+G(d)& $1.486\pm 0.005$ & $3.348\pm 0.024$ & $4.084\pm 0.012$ \\
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\end{squeezetable}
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%%% %%% %%% %%%
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Figure \ref{fig:D2h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_tddft_D2h} and \ref{tab:D2h}. We see that the SF-TD-DFT vertical energies are rather close to the TBE one for the doubly excited state {\Ag}. Indeed, as presented later in subsection \ref{sec:TBE}, SF methods are able to describe effectively states with strong multiconfigurational character. The same observation can be done for the SF-ADC values but with much better results for the two other states. Again we can notice that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the values. For the multireference methods we see that using the smallest active space do not provide a good description of the {\sBoneg} state and in the case of CASSCF(4,4), as previously said, we even have {\sBoneg} state above the {\Ag} state. For the CC methods taking into account the quadruple excitations is necessary to have a good description of the doubly excited state {\Ag}.
|
||||
Figure \ref{fig:D2h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_tddft_D2h} and \ref{tab:D2h}. We see that the SF-TD-DFT vertical energies are rather close to the TBE one for the doubly excited state {\twoAg}. Indeed, as presented later in subsection \ref{sec:TBE}, SF methods are able to describe effectively states with strong multiconfigurational character. The same observation can be done for the SF-ADC values but with much better results for the two other states. Again we can notice that the SF-ADC(2)-x and SF-ADC(3) schemes do not improve the values. For the multireference methods we see that using the smallest active space do not provide a good description of the {\sBoneg} state and in the case of CASSCF(4,4), as previously said, we even have {\sBoneg} state above the {\twoAg} state. For the CC methods taking into account the quadruple excitations is necessary to have a good description of the doubly excited state {\twoAg}.
|
||||
|
||||
%%% FIGURE III %%%
|
||||
\begin{figure*}
|
||||
%width=0.8\linewidth
|
||||
\includegraphics[scale=0.5]{D2h.pdf}
|
||||
\caption{Vertical energies of the {\tBoneg}, {\sBoneg} and {\Ag} states for the {\Dtwo} geometry using the aug-cc-pVTZ basis. Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multireference, CC, and TBE values, respectively.}
|
||||
\caption{Vertical energies of the {\tBoneg}, {\sBoneg} and {\twoAg} states for the {\Dtwo} geometry using the aug-cc-pVTZ basis. Purple, orange, green, blue and black lines correspond to the SF-TD-DFT, SF-ADC, multireference, CC, and TBE values, respectively.}
|
||||
\label{fig:D2h}
|
||||
\end{figure*}
|
||||
%%% %%% %%% %%%
|
||||
@ -641,17 +642,17 @@ Figure \ref{fig:D2h} shows the vertical energies of the studied excited states d
|
||||
|
||||
\subsubsection{{\Dfour} geometry}
|
||||
\label{sec:D4h}
|
||||
Table \ref{tab:sf_D4h} shows vertical energies obtained with spin-flip methods and Table \ref{tab:D4h} vertical energies obtained with standard methods. As for the previous geometry we start with the SF-TD-DFT results with hybrid functionals. We can first notice that for the B3LYP and the PBE0 functionals we have a wrong ordering of the first triplet state {\Atwog} and the ground state $\text{X}\,{}^1B_{1g}$. We retrieve the good ordering with the BH\&HLYP functional, so adding exact exchange to the functional allows us to have the right ordering between these two states. For the B3LYP and the PBE0 functionals we have that the energy differences for each states and for all bases are small with \SIrange{0.004}{0.007}{\eV} for the triplet state {\Atwog}. We have \SIrange{0.015}{0.021}{\eV} of energy difference for the {\Aoneg} state through all bases, we can notice that this state is around \SI{0.13}{\eV} (considering all bases) higher with the PBE0 functional. We can make the same observation for the {\Btwog} state for the B3LYP and the PBE0 functionals, indeed we have small energy differences for all bases and the state is around \SIrange{0.14}{0.15}{\eV} for the PBE0 functional. For the BH\&HLYP functional the {\Aoneg} and {\Btwog} states are higher in energy than for the two other hybrid functionals with about \SIrange{0.65}{0.69}{\eV} higher for the {\Aoneg} state and \SIrange{0.75}{0.77}{\eV} for the {\Btwog} state compared to the PBE0 functional. Then, we have the RSH functionals CAM-B3LYP, $\omega$B97X-V and LC-$\omega$PBE08. For these functionals the vertical energies are similar for the {\Aoneg} and {\Btwog} states with a maximum energy difference of \SIrange{0.01}{0.02}{\eV} for the{\Aoneg} state and \SIrange{0.005}{0.009}{\eV} for the {\Btwog} state considering all bases. The maximum energy difference for the triplet state is larger with \SIrange{0.047}{0.057}{\eV} for all bases. Note that the vertical energies obtained with the RSH functionals are close to the PBE0 ones except that we have the right ordering between the triplet state {\Atwog} and the ground state $\text{X}\,{}^1B_{1g}$. The M06-2X results are closer to the BH\&HLYP ones but the M06-2X vertical energies are always higher than the BH\&HLYP ones. We can notice that the M06-2X energies for the {\Aoneg} state are close to the BH\&HLYP energies for the {\Btwog} state. For these two states, when we compared the results obtained with the M06-2X and the BH\&HLYP functionals, we have an energy difference of \SIrange{0.16}{0.17}{\eV} for the {\Aoneg} state and \SIrange{0.17}{0.18}{\eV} for the {\Btwog} state considering all bases. For the triplet state {\Atwog} the energy differences are smaller with \SIrange{0.03}{0.04}{\eV} for all bases. The M11 vertical energies are very close to the M06-2X ones for the triplet state with a maximum energy difference of \SI{0.003}{\eV} considering all bases, and are closer to the BH\&HLYP results for the two other states with \SIrange{0.06}{0.07}{\eV} and \SIrange{0.07}{0.08}{\eV} of energy difference for the {\Aoneg} and {\Btwog} states, respectively. Then we discuss the various ADC scheme results, note that we were not able to obtain the vertical energies with the aug-cc-pVQZ basis due to computational resources. For the ADC(2)-s scheme we can see that the energy difference for the triplet state are smaller than for the two other states, indeed we have an energy difference of \SI{0.09}{\eV} for the triplet state whereas we have \SI{0.15}{\eV} and \SI{0.25}{eV} for the {\Aoneg} and {\Btwog} states, again when considering all bases. The energy difference for each state and through the bases are similar for the two other ADC schemes. We can notice a large variation of the vertical energies for the {\Aoneg} state between ADC(2)-s and ADC(2)-x with around \SIrange{0.52}{0.58}{\eV} through all bases. The ADC(3) vertical energies are very similar to the ADC(2) ones for the {\Btwog} state with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases, whereas we have an energy difference of \SIrange{0.04}{0.11}{\eV} and \SIrange{0.17}{0.22}{\eV} for the {\Aoneg} and {\Btwog} states, respectively.
|
||||
Table \ref{tab:sf_D4h} shows vertical energies obtained with spin-flip methods and Table \ref{tab:D4h} vertical energies obtained with standard methods. As for the previous geometry we start with the SF-TD-DFT results with hybrid functionals. We can first notice that for the B3LYP and the PBE0 functionals we have a wrong ordering of the first triplet state {\Atwog} and the ground state {\sBoneg}. We retrieve the good ordering with the BH\&HLYP functional, so adding exact exchange to the functional allows us to have the right ordering between these two states. For the B3LYP and the PBE0 functionals we have that the energy differences for each states and for all bases are small with \SIrange{0.004}{0.007}{\eV} for the triplet state {\Atwog}. We have \SIrange{0.015}{0.021}{\eV} of energy difference for the {\Aoneg} state through all bases, we can notice that this state is around \SI{0.13}{\eV} (considering all bases) higher with the PBE0 functional. We can make the same observation for the {\Btwog} state for the B3LYP and the PBE0 functionals, indeed we have small energy differences for all bases and the state is around \SIrange{0.14}{0.15}{\eV} for the PBE0 functional. For the BH\&HLYP functional the {\Aoneg} and {\Btwog} states are higher in energy than for the two other hybrid functionals with about \SIrange{0.65}{0.69}{\eV} higher for the {\Aoneg} state and \SIrange{0.75}{0.77}{\eV} for the {\Btwog} state compared to the PBE0 functional. Then, we have the RSH functionals CAM-B3LYP, $\omega$B97X-V and LC-$\omega$PBE08. For these functionals the vertical energies are similar for the {\Aoneg} and {\Btwog} states with a maximum energy difference of \SIrange{0.01}{0.02}{\eV} for the{\Aoneg} state and \SIrange{0.005}{0.009}{\eV} for the {\Btwog} state considering all bases. The maximum energy difference for the triplet state is larger with \SIrange{0.047}{0.057}{\eV} for all bases. Note that the vertical energies obtained with the RSH functionals are close to the PBE0 ones except that we have the right ordering between the triplet state {\Atwog} and the ground state {\sBoneg}. The M06-2X results are closer to the BH\&HLYP ones but the M06-2X vertical energies are always higher than the BH\&HLYP ones. We can notice that the M06-2X energies for the {\Aoneg} state are close to the BH\&HLYP energies for the {\Btwog} state. For these two states, when we compared the results obtained with the M06-2X and the BH\&HLYP functionals, we have an energy difference of \SIrange{0.16}{0.17}{\eV} for the {\Aoneg} state and \SIrange{0.17}{0.18}{\eV} for the {\Btwog} state considering all bases. For the triplet state {\Atwog} the energy differences are smaller with \SIrange{0.03}{0.04}{\eV} for all bases. The M11 vertical energies are very close to the M06-2X ones for the triplet state with a maximum energy difference of \SI{0.003}{\eV} considering all bases, and are closer to the BH\&HLYP results for the two other states with \SIrange{0.06}{0.07}{\eV} and \SIrange{0.07}{0.08}{\eV} of energy difference for the {\Aoneg} and {\Btwog} states, respectively. Then we discuss the various ADC scheme results, note that we were not able to obtain the vertical energies with the aug-cc-pVQZ basis due to computational resources. For the ADC(2)-s scheme we can see that the energy difference for the triplet state are smaller than for the two other states, indeed we have an energy difference of \SI{0.09}{\eV} for the triplet state whereas we have \SI{0.15}{\eV} and \SI{0.25}{eV} for the {\Aoneg} and {\Btwog} states, again when considering all bases. The energy difference for each state and through the bases are similar for the two other ADC schemes. We can notice a large variation of the vertical energies for the {\Aoneg} state between ADC(2)-s and ADC(2)-x with around \SIrange{0.52}{0.58}{\eV} through all bases. The ADC(3) vertical energies are very similar to the ADC(2) ones for the {\Btwog} state with an energy difference of \SIrange{0.01}{0.02}{\eV} for all bases, whereas we have an energy difference of \SIrange{0.04}{0.11}{\eV} and \SIrange{0.17}{0.22}{\eV} for the {\Aoneg} and {\Btwog} states, respectively.
|
||||
|
||||
Now we look at Table \ref{tab:D4h} for vertical energies obtained with standard methods, we have first the various CC methods, for all the CC methods we were only able to reach the aug-cc-pVTZ basis.
|
||||
%For the CCSD method we cannot obtain the vertical energy for the $1\,{}^1B_{2g}$ state, we can notice a small energy variation through the bases for the triplet state with around 0.06 eV. The energy variation is larger for the $2\,{}^1A_{1g}$ state with 0.20 eV.
|
||||
Again, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. Note that for CC we started from a restricted Hartree-Fock (RHF) reference and in that case the ground state is the {\Aoneg}state, then the $X\,{}^1B_{1g}$ state is the single deexcitation and the{\Btwog} state is the double excitation from our ground state. For the CC3 method we do not have the vertical energies for the triplet state {\Atwog}. Considering all bases for the {\Aoneg}and {\Btwog} states we have an energy difference of about \SI{0.15}{\eV} and \SI{0.12}{\eV}, respectively. The CCSDT energies are close to the CC3 ones for the {\Aoneg} state with an energy difference of around \SIrange{0.03}{0.06}{\eV} considering all bases. For the{\Btwog} state the energy difference between the CC3 and the CCSDT values is larger with \SIrange{0.18}{0.27}{\eV}. We can make a similar observation between the CC4 and the CCSDTQ values, for the {\Aoneg} state we have an energy difference of about \SI{0.01}{\eV} and this time we have smaller energy difference for the {\Btwog} with \SI{0.01}{\eV}. Then we discuss the multireference results and this time we were able to reach the aug-cc-pVQZ basis. Again we consider two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals. If we compare the CASSCF(4,4) and CASPT2(4,4) values we can see large energy differences for the {\Aoneg} and {\Btwog} states, for the {\Aoneg} state we have an energy difference of about \SIrange{0.67}{0.74}{\eV} and \SIrange{1.65}{1.81}{\eV} for the {\Btwog}state. The energy difference is smaller for the triplet state with \SIrange{0.27}{0.31}{\eV}, we can notice that all the CASSCF(4,4) vertical energies are higher than the CASPT2(4,4) ones. Then we have the NEVPT2(4,4) methods (SC-NEVPT2(4,4) and PC-NEVPT2(4,4)) for which the vertical energies are quite similar, however we can notice that we have a different ordering of the {\Aoneg} and {\Btwog} states with {\Btwog} higher in energy than {\Aoneg} for the two NEVPT2(4,4) methods. Then we have the results for the same methods but with a larger active space. For the CASSCF(12,12) we have similar values for the triplet state with energy difference of about \SI{0.06}{\eV} for all bases but larger energy difference for the {\Aoneg} state with around \SIrange{0.28}{0.29}{\eV} and \SIrange{0.79}{0.81}{\eV} for the {\Btwog} state. Note that from CASSCF(4,4) to CASSCF(12,12) every vertical energy is smaller. We can make the contrary observation for the CASPT2 method where from CASPT2(4,4) to CASPT2(12,12) every vertical energy is larger. For the CASPT2(12,12) method we have similar values for the triplet state and for the {\Aoneg} state, considering all bases, with an energy difference of around \SIrange{0.05}{0.06}{\eV} and \SIrange{0.02}{0.05}{\eV} respectively. The energy difference is larger for the {\Btwog} state with about \SIrange{0.27}{0.29}{\eV}. Next we have the NEVPT2 methods, the first observation that we can make is that by increasing the size of the active space we retrieve the right ordering of the {\Aoneg} and {\Btwog} states. Again, every vertical energy is higher in the NEVPT2(12,12) case than for the NEVPT2(4,4) one.
|
||||
Again, we have computed the vertical energies at the CCSDT and CCSDTQ level as well as their approximation the CC3 and CC4 methods, respectively. Note that for CC we started from a restricted Hartree-Fock (RHF) reference and in that case the ground state is the {\Aoneg}state, then the {\sBoneg} state is the single deexcitation and the {\Btwog} state is the double excitation from our ground state. For the CC3 method we do not have the vertical energies for the triplet state {\Atwog}. Considering all bases for the {\Aoneg}and {\Btwog} states we have an energy difference of about \SI{0.15}{\eV} and \SI{0.12}{\eV}, respectively. The CCSDT energies are close to the CC3 ones for the {\Aoneg} state with an energy difference of around \SIrange{0.03}{0.06}{\eV} considering all bases. For the{\Btwog} state the energy difference between the CC3 and the CCSDT values is larger with \SIrange{0.18}{0.27}{\eV}. We can make a similar observation between the CC4 and the CCSDTQ values, for the {\Aoneg} state we have an energy difference of about \SI{0.01}{\eV} and this time we have smaller energy difference for the {\Btwog} with \SI{0.01}{\eV}. Then we discuss the multireference results and this time we were able to reach the aug-cc-pVQZ basis. Again we consider two different active spaces, four electrons in four orbitals and twelve electrons in twelve orbitals. If we compare the CASSCF(4,4) and CASPT2(4,4) values we can see large energy differences for the {\Aoneg} and {\Btwog} states, for the {\Aoneg} state we have an energy difference of about \SIrange{0.67}{0.74}{\eV} and \SIrange{1.65}{1.81}{\eV} for the {\Btwog} state. The energy difference is smaller for the triplet state with \SIrange{0.27}{0.31}{\eV}, we can notice that all the CASSCF(4,4) vertical energies are higher than the CASPT2(4,4) ones. Then we have the NEVPT2(4,4) methods (SC-NEVPT2(4,4) and PC-NEVPT2(4,4)) for which the vertical energies are quite similar, however we can notice that we have a different ordering of the {\Aoneg} and {\Btwog} states with {\Btwog} higher in energy than {\Aoneg} for the two NEVPT2(4,4) methods. Then we have the results for the same methods but with a larger active space. For the CASSCF(12,12) we have similar values for the triplet state with energy difference of about \SI{0.06}{\eV} for all bases but larger energy difference for the {\Aoneg} state with around \SIrange{0.28}{0.29}{\eV} and \SIrange{0.79}{0.81}{\eV} for the {\Btwog} state. Note that from CASSCF(4,4) to CASSCF(12,12) every vertical energy is smaller. We can make the contrary observation for the CASPT2 method where from CASPT2(4,4) to CASPT2(12,12) every vertical energy is larger. For the CASPT2(12,12) method we have similar values for the triplet state and for the {\Aoneg} state, considering all bases, with an energy difference of around \SIrange{0.05}{0.06}{\eV} and \SIrange{0.02}{0.05}{\eV} respectively. The energy difference is larger for the {\Btwog} state with about \SIrange{0.27}{0.29}{\eV}. Next we have the NEVPT2 methods, the first observation that we can make is that by increasing the size of the active space we retrieve the right ordering of the {\Aoneg} and {\Btwog} states. Again, every vertical energy is higher in the NEVPT2(12,12) case than for the NEVPT2(4,4) one.
|
||||
|
||||
%%% TABLE VI %%%
|
||||
\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{
|
||||
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the \Dfour square-planar equilibrium geometry of the {\Atwog} state.
|
||||
Standard vertical excitation energies (with respect to the singlet {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.
|
||||
\label{tab:sf_D4h}}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{llrrr}
|
||||
@ -716,7 +717,7 @@ SF-ADC(3) & 6-31+G(d) & $0.123$ & $1.650$ & $2.078$ \\
|
||||
\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{
|
||||
Standard vertical excitation energies (with respect to the singlet $\text{X}\,{}^1B_{1g}$ ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.
|
||||
Standard vertical excitation energies (with respect to the singlet {\sBoneg} ground state) of the {\Atwog}, {\Aoneg}, and {\Btwog} states of CBD at the {\Dfour} square-planar equilibrium geometry of the {\Atwog} state.
|
||||
\label{tab:D4h}}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{llrrr}
|
||||
@ -791,7 +792,7 @@ CIPSI & 6-31+G(d) & $0.2010\pm 0.0030$ & $1.602\pm 0.007$ & $2.13\pm 0.04$ \\
|
||||
\end{squeezetable}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
Figure \ref{fig:D4h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_D4h} and \ref{tab:D4h}. We see that all methods exhibit larger variations of the vertical energies due to the high symmetry of the \Dfour structure. For the triplet state most of the methods used are able to give a vertical energy close to the TBE one, we can although see that CASSCF, with the two different active spaces, show larger energy error than most of the DFT functionals used. Then for the, strongly multiconfigurational character, {\Aoneg} state we have a good description by the CC and multireference methods with the largest active space, except for CASSCF. The SF-ADC(2) and SF-ADC(3) schemes are also able to provide a good description of the {\Aoneg} state and even for the {\Btwog} we see that SF-ADC(2)-x give a worst result than SF-ADC(2). Multiconfigurational methods with the smallest active space do not demonstrate a good description of the two singlet excited states especially for the CASSCF method and for the the NEVPT2 methods that give the vertical energy of the {\Btwog} state below the {\Aoneg} one. Note that CASPT2 improve a lot the description of all the states compared to CASSCF. The various TD-DFT functionals are not able to describe correctly the two singlet excited states.
|
||||
Figure \ref{fig:D4h} shows the vertical energies of the studied excited states described in Tables \ref{tab:sf_D4h} and \ref{tab:D4h}. We see that all methods exhibit larger variations of the vertical energies due to the high symmetry of the {\Dfour} structure. For the triplet state most of the methods used are able to give a vertical energy close to the TBE one, we can although see that CASSCF, with the two different active spaces, show larger energy error than most of the DFT functionals used. Then for the, strongly multiconfigurational character, {\Aoneg} state we have a good description by the CC and multireference methods with the largest active space, except for CASSCF. The SF-ADC(2) and SF-ADC(3) schemes are also able to provide a good description of the {\Aoneg} state and even for the {\Btwog} we see that SF-ADC(2)-x give a worst result than SF-ADC(2). Multiconfigurational methods with the smallest active space do not demonstrate a good description of the two singlet excited states especially for the CASSCF method and for the the NEVPT2 methods that give the vertical energy of the {\Btwog} state below the {\Aoneg} one. Note that CASPT2 improve a lot the description of all the states compared to CASSCF. The various TD-DFT functionals are not able to describe correctly the two singlet excited states.
|
||||
|
||||
%%% FIGURE IV %%%
|
||||
\begin{figure*}
|
||||
@ -810,7 +811,7 @@ Then we look at the vertical energy errors for the {\Dtwo} structure. First we c
|
||||
|
||||
For the {\sBoneg} state of the {\Dtwo} structure we see that all the xc-functional underestimate the vertical excitation energy with energy differences of about \SIrange{0.35}{0.93}{\eV}. The ADC values are much closer to the TBE with energy differences around \SIrange{0.03}{0.09}{\eV}. Obviously, the CC vertical energies are close to the TBE one with around or less than \SI{0.01}{\eV} of energy difference. For the CASSCF(4,4) vertical energy we have a large difference of around \SI{1.42}{\eV} compared to the TBE value due to the lack of dynamical correlation in the CASSCF method. As previously seen the CAPT2(4,4) method correct this and we obtain a value of \SI{0.20}{\eV}. The others multireference methods in this active space give energy differences of around \SIrange{0.55}{0.76}{\eV} compared the the TBE reference. For the largest active space with twelve electrons in twelve orbitals we have an improvement of the vertical energies with \SI{0.72}{eV} of energy difference for the CASSCF(12,12) method and around 0.06 eV for the others multiconfigurational methods.
|
||||
|
||||
Then, for the {\Ag} state we obtain closer results of the SF-TD-DFT methods to the TBE than in the case of the {\sBoneg} state. Indeed, we have an energy difference of about \SIrange{0.01}{0.34}{\eV} for the {\Ag} state whereas we have \SIrange{0.35}{0.93}{\eV} for the {\sBoneg} state. The ADC schemes give the same error to the TBE value than for the other singlet state with \SI{0.02}{\eV} for the ADC(2) scheme and \SI{0.07}{\eV} for the ADC(3) one. The ADC(2)-x scheme provides a larger error with \SI{0.45}{\eV} of energy difference. Here, the CC methods manifest more variations with \SI{0.63}{\eV} for the CC3 value and \SI{0.28}{\eV} for the CCSDT compared to the TBE values. The CC4 method provides a small error with less than 0.01 eV of energy difference. The multiconfigurational methods globally give smaller error than for the other singlet state with, for the two active spaces, \SIrange{0.03}{0.12}{\eV} compared to the TBE value. We can notice that CC3 and CCSDT provide larger energy errors for the {\Ag} state than for the {\sBoneg} state, this is due to the strong multiconfigurational character of the {\Ag} state whereas the {\sBoneg} state has a very weak multiconfigurational character. It is interesting to see that SF-TD-DFT and SF-ADC methods give better results compared to the TBE than CC3 and even CCSDT meaning that spin-flip methods are able to describe double excited states. Note that multireference methods obviously give better results too for the {\Ag} state.
|
||||
Then, for the {\twoAg} state we obtain closer results of the SF-TD-DFT methods to the TBE than in the case of the {\sBoneg} state. Indeed, we have an energy difference of about \SIrange{0.01}{0.34}{\eV} for the {\twoAg} state whereas we have \SIrange{0.35}{0.93}{\eV} for the {\sBoneg} state. The ADC schemes give the same error to the TBE value than for the other singlet state with \SI{0.02}{\eV} for the ADC(2) scheme and \SI{0.07}{\eV} for the ADC(3) one. The ADC(2)-x scheme provides a larger error with \SI{0.45}{\eV} of energy difference. Here, the CC methods manifest more variations with \SI{0.63}{\eV} for the CC3 value and \SI{0.28}{\eV} for the CCSDT compared to the TBE values. The CC4 method provides a small error with less than 0.01 eV of energy difference. The multiconfigurational methods globally give smaller error than for the other singlet state with, for the two active spaces, \SIrange{0.03}{0.12}{\eV} compared to the TBE value. We can notice that CC3 and CCSDT provide larger energy errors for the {\twoAg} state than for the {\sBoneg} state, this is due to the strong multiconfigurational character of the {\twoAg} state whereas the {\sBoneg} state has a very weak multiconfigurational character. It is interesting to see that SF-TD-DFT and SF-ADC methods give better results compared to the TBE than CC3 and even CCSDT meaning that spin-flip methods are able to describe double excited states. Note that multireference methods obviously give better results too for the {\twoAg} state.
|
||||
|
||||
Finally we look at the vertical energy errors for the \Dfour structure. First, we consider the {\Atwog} state, the SF-TD-DFT methods give errors of about \SIrange{0.07}{1.6}{\eV} where the largest energy differences are provided by the hybrid functionals. The ADC schemes give similar error with around \SIrange{0.06}{1.1}{\eV} of energy difference. For the CC methods we have an energy error of \SI{0.06}{\eV} for CCSD and less than \SI{0.01}{\eV} for CCSDT. Then for the multireference methods with the four by four active space we have for CASSCF(4,4) \SI{0.29}{\eV} of error and \SI{0.02}{\eV} for CASPT2(4,4), again CASPT2 demonstrates its improvement compared to CASSCF. The other methods provide energy differences of about \SIrange{0.12}{0.13}{\eV}. A larger active space shows again an improvement with \SI{0.23}{\eV} of error for CASSCF(12,12) and around \SIrange{0.01}{0.04}{\eV} for the other multireference methods. CIPSI provides similar error with \SI{0.02}{\eV}. Then, we look at the {\Aoneg} state where the SF-TD-DFT shows large variations of error depending on the functionals, the energy error is about \SIrange{0.10}{1.03}{\eV}. The ADC schemes give better errors with around \SIrange{0.07}{0.41}{\eV} and where again the ADC(2)-x does not improve the ADC(2)-s energy but also gives worse results. For the CC methods we have energy errors of about \SIrange{0.10}{0.16}{\eV} and CC4 provides really close energy to the TBE one with \SI{0.01}{\eV} of error. For the multireference methods we globally have an improvement of the energies from the four by four to the twelve by twelve active space with errors of \SIrange{0.01}{0.73}{\eV} and \SIrange{0.02}{0.44}{\eV} respectively with the largest errors coming from the CASSCF method. Lastly, we look at the {\Btwog} state where we have globally larger errors. The SF-TD-DFT exhibits errors of \SIrange{0.43}{1.50}{\eV} whereas ADC schemes give errors of \SIrange{0.18}{0.30}{\eV}. CC3 and CCSDT provide energy differences of \SIrange{0.50}{0.69}{\eV} and the CC4 shows again close energy to the CCSDTQ TBE energy with \SI{0.01}{\eV} of error. The multireference methods give energy differences of \SIrange{0.38}{1.39}{\eV} and \SIrange{0.11}{0.60}{\eV} for the four by four and twelve by twelve active spaces respectively. We can notice that we have larger variations for the vertical energies of the square CBD than for the rectangular one. This can be explained by the fact that because of the degeneracy in the {\Dfour} structure it leads to strong multiconfigurational character states where single reference methods are unreliable. We can also see that for the CC methods we have a better description of the {\Aoneg} state than the {\Btwog} state, this can be related, as previously said in Subsubsection \ref{sec:D4h}, to the fact that {\Btwog} corresponds to a double excitation from the reference state. To obtain an improved description of the {\Btwog} state we have to include quadruples. At the end of Table \ref{tab:TBE} we show some literature results obtain from Ref.~\onlinecite{lefrancois_2015,manohar_2008} where the cc-pVTZ basis is used. The SF-ADC(2)-s, SF-ADC(2)-x and SF-ADC(3)results are presented and are consistent with our results with the exact same schemes but with the aug-cc-pVTZ basis.
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|
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@ -885,7 +886,7 @@ Finally we look at the vertical energy errors for the \Dfour structure. First, w
|
||||
%\begin{tabular}{*{1}{*{8}{l}}}
|
||||
& &\mc{3}{c}{{\Dtwo} excitation energies (eV)} & \mc{3}{c}{{\Dfour} excitation energies (eV)} \\
|
||||
\cline{3-5} \cline{6-8}
|
||||
Method & AB (\si{\kcalmol}) & {\tBoneg}(99\%) & {\sBoneg}(95\%)& {\Ag}(1\%) & $1\,{}^3A_{2g} $ & {\Aoneg} & {\Btwog} \\
|
||||
Method & AB (\si{\kcalmol}) & {\tBoneg}(99\%) & {\sBoneg}(95\%)& {\twoAg}(1\%) & {\Atwog} & {\Aoneg} & {\Btwog} \\
|
||||
\hline
|
||||
SF-TD-B3LYP & $10.41$ & $0.241$ & $-0.926$ & $-0.161$ & $-0.164$ & $-1.028$ & $-1.501$ \\
|
||||
SF-TD-PBE0 & $8.95$ & $0.220$ & $-0.829$ & $-0.068$ & $-0.163$ & $-0.903$ & $-1.357$ \\
|
||||
@ -950,7 +951,7 @@ In order to provide a benchmark of the AB and vertical energies we used coupled-
|
||||
|
||||
With SF-TD-DFT the quality of the results are, of course, dependent on the functional but for the doubly excited states we have solid results. In SF-ADC we have very good results compared to the TBEs even for the doubly excited states, nevertheless the ADC(2)-x scheme give almost systematically worse results than the ADC(2)-s ones and using the ADC(3) scheme does not always provide better values.
|
||||
|
||||
The description of the excited states of the {\Dtwo} structure give rise to good agreement between the single reference and multiconfigurational methods due to the large T1 percentage of the first two excited states. When this percentage is much smaller as in the case of the doubly excited state {\Ag} the spin-flip methods show very good results within the ADC framework but even at the TD-DFT level spin-flip display good results compared to the TBE value. As said in the discussion, for the {\Dfour} geometry, the description of excited states is harder because of the strong multiconfigurational character where SF-TD-DFT can present more than 1 eV of error compared to the TBE. However, SF-ADC can show error of around \SIrange{0.1}{0.2}{\eV} which can be better than the multiconfigurational methods results.
|
||||
The description of the excited states of the {\Dtwo} structure give rise to good agreement between the single reference and multiconfigurational methods due to the large T1 percentage of the first two excited states. When this percentage is much smaller as in the case of the doubly excited state {\twoAg} the spin-flip methods show very good results within the ADC framework but even at the TD-DFT level spin-flip display good results compared to the TBE value. As said in the discussion, for the {\Dfour} geometry, the description of excited states is harder because of the strong multiconfigurational character where SF-TD-DFT can present more than 1 eV of error compared to the TBE. However, SF-ADC can show error of around \SIrange{0.1}{0.2}{\eV} which can be better than the multiconfigurational methods results.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\acknowledgements{
|
||||
|
||||
@ -1,2 +1 @@
|
||||
\relax
|
||||
\gdef \@abspage@last{1}
|
||||
|
||||
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@ -62,7 +62,7 @@
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||||
|
||||
%left%
|
||||
\draw (-2.5,-1.5) -- (-3.5,-1.5);
|
||||
\node[] at (-4.1,-1.5) {$^1$A$_{g}$};
|
||||
\node[] at (-4.1,-1.5) {$1{}^1$A$_{g}$};
|
||||
|
||||
\draw[red] (-2.5,9.5) -- (-3.5,9.5);
|
||||
\node[red] at (-4.1,9.5) {$1~^3$B$_{1g}$};
|
||||
@ -70,14 +70,14 @@
|
||||
|
||||
%right%
|
||||
\draw (2.5,-1.5) -- (3.5,-1.5);
|
||||
\node[] at (4.1,-1.5) {$^1$A$_{g}$};
|
||||
\node[] at (4.1,-1.5) {$1{}^1$A$_{g}$};
|
||||
|
||||
\draw[red] (2.5,9.5) -- (3.5,9.5);
|
||||
\node[red] at (4.1,9.5) {$1~^3$B$_{1g}$};
|
||||
|
||||
%above%
|
||||
\draw (-0.5,1.5) -- (0.5,1.5);
|
||||
\node[] at (1,1.4) {$^1$B$_{1g}$};
|
||||
\node[] at (1,1.4) {$1{}^1$B$_{1g}$};
|
||||
|
||||
\draw[red] (-0.5,3.5) -- (0.5,3.5);
|
||||
\node[red] at (1.1,3.5) {$1~^3$A$_{2g}$};
|
||||
|
||||
Loading…
Reference in New Issue
Block a user